Recovery of disrupted airline operations using k-Maximum Matching in graphs

By Berge's theorem, finding a maximum matching in a graph relies on the use of augmenting paths. When no further constraint is added, Edmonds' algorithm allows to compute a maximum matching in polynomial time by sequentially augmenting such paths. Motivated by applications in the scheduling of airline operations, we consider a similar problem where only paths of bounded length can be augmented. Precisely, let k≥1 be an odd integer, and M a matching of a graph G. What is the maximum size of a matching that can be obtained from M by using only augmenting paths of length at most k? We first prove that this problem can be solved in polynomial time for k≤3 in any graph and that it is NP-complete for any fixed k≥5 in the class of planar bipartite graphs of degree at most 3 and arbitrarily large girth. We then prove that this problem is in P, for any k, in several subclasses of trees such as caterpillars or trees with all vertices of degree at least 3 “far appart”. Moreover, this problem can be solved in time O(n) in the class of n-node trees when k and the maximum degree are fixed parameters. Finally, we consider a more constrained problem where only paths of length exactly k can be augmented. We prove that this latter problem becomes NP-complete for any fixed k≥3 and in trees when k is part of the input.